Multivalued forbidden numbers of two-rowed configurations -- the missing cases

TL;DR

This paper advances extremal combinatorics by determining forbidden matrix configurations for 3-valued matrices with two rows, especially focusing on missing cases and asymptotic behaviors for specific configurations.

Contribution

It completes the classification of forbidden configurations for 2-rowed matrices over a 3-valued alphabet, providing asymptotic and exact results for previously unresolved cases.

Findings

Asymptotic behavior of forb(m,3,p·K2) - forb(m,3,p·I2) for p>3 determined.

Exact values of forb(m,3,F) computed for many 2-rowed 2-matrices.

Systematic extension of prior work on forbidden configurations in 2-matrices.

Abstract

The present paper considers extremal combinatorics questions in the language of matrices. An $s$-matrix is a matrix with entries in $\{0,1,\ldots, s-1\}$. An $s$-matrix is simple if it has no repeated columns. A matrix $F$ is a configuration in a matrix $A$, denoted $F\prec A$, if it is a row/column permutation of a submatrix of $A$. $\text{Avoid}(m,s,F)$ is the set of $m$-rowed, simple $s$-matrices not containing a configuration of $F$ and $\text{forb}(m,s, F)=\max\{|A|\colon A \in \text{Avoid}(m,s,F)\}$. Dillon and Sali initiated the systematic study of $\text{forb}(m,s, F)$ for $2$-matrices $F$, and computed $\text{forb}(m,s, F)$ for all 2-rowed $F$ when $s>3$. In this paper we tackle the remaining cases when $s=3$. In particular, we determine the asymptotics of $\text{forb}(m,3,p\cdot K_2)-\text{forb}(m,3,p\cdot I_2)$ for $p>3$, where $K_2$ is the $2\times 4$ simple $2$-matrix and $I_2$ is the $2\times 2$ identity matrix, as well as the exact values of $\text{forb}(m,3,F)$ for many 2-rowed $2$-matrices $F$.